Remark 2.4 (Pythagorean identity) 
A classical identity relating the eigenvalue and eigenfunction errors (see, e.g.,
[1]}) is
\(\Vert  u_j - \hat{u}_j^h \Vert _E^2 = \lambda _j \Vert  u_j - \hat{u}_j^h \Vert ^2_{\Omega } + \hat{\lambda }_j^h - \lambda _j,\) 
